Determine Dumbbell Distribution: Solving for 7.1 kg Weights Using Averages

Sebastian has 17 dumbbells that weigh on average $5.22$ kg.

3 of the dumbbells weigh 4.5 kg, 4 dumbbells weigh 5.2 kg, and the rest weigh 7.1 kg or 3.8 kg.

How many dumbbells weighing 7.1 kg does Sebastian have?

To solve for the number of dumbbells weighing 7.1 kg, we will leverage the weighted average given by the problem:

• Calculate weights of known dumbbells:
  • Total weight of 3 dumbbells at 4.5 kg each: $3 \times 4.5 = 13.5$ kg
  • Total weight of 4 dumbbells at 5.2 kg each: $4 \times 5.2 = 20.8$ kg
• Let $x$ be the number of dumbbells weighing 7.1 kg. Therefore, the remaining $10 - x$ dumbbells weigh 3.8 kg each.
• Calculate total weight: $13.5 + 20.8 + 7.1x + 3.8(10 - x) = 5.22 \times 17$ Simplifying the right, we have: $13.5 + 20.8 + 7.1x + 38 - 3.8x = 88.74$ $7.1x - 3.8x + 13.5 + 20.8 + 38 = 88.74$ $3.3x + 72.3 = 88.74$ $3.3x = 88.74 - 72.3$ $3.3x = 16.44$ $x = \frac{16.44}{3.3} = 5$

Therefore, the number of dumbbells weighing 7.1 kg is $5$.